where \({\cal T}_k (\cdot )\)  represents a thresholding operator that is used to produce a \(k\) -sparse vector, \(\lambda \)  denotes the stepsize and \(d\)  is a search direction at the current iterate \(x^p\) .
Throughout the paper, a vector \(x\)  is said to be \(k\) -sparse if \({x}_0 \le k\) .
Several thresholding operators are widely used in the literature, such as the hard thresholding [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, soft thresholding [7]}, [8]}, [9]}, [10]}, and optimal \(k\) -thresholding [11]}, [12]}.
The steepest descent direction [13]}, [1]}, [15]}, [2]}, [5]}, [18]} and Newton-type direction [19]}, [20]}, [21]}, [22]} are two search directions that are used by many researchers.
